Life Tables: Mortality Rates, Survival Functions, and Notation
Master life table notation, mortality rates, and survival functions essential for Exam LTAM.
The Life Table Framework
The life table is the foundational data structure in life contingencies, providing a systematic way to describe the mortality experience of a population. It tracks a hypothetical cohort of lives from birth (or some starting age) through the ages, recording how many survive to each age and how many die within each age interval. From this simple structure, all the probabilities needed for life insurance and annuity calculations can be derived.
The core functions of the life table are interconnected. The radix l_0 is the initial cohort size (often set to 100,000 for convenience). At each subsequent age x, l_x represents the number of survivors. The number of deaths between ages x and x+1 is d_x = l_x - l_{x+1}. From these counts, we derive the fundamental probability: q_x = d_x/l_x, the probability that a person aged exactly x dies before reaching age x+1. The complement p_x = 1 - q_x is the probability of surviving from age x to age x+1.
Multi-Year Probabilities
Insurance products span multiple years, so we need multi-year survival and mortality probabilities. The notation t_p_x represents the probability that a person aged x survives for at least t more years: t_p_x = l_{x+t}/l_x. This can also be expressed as a product of one-year survival probabilities: t_p_x = p_x * p_{x+1} * p_{x+2} * ... * p_{x+t-1}.
The multi-year mortality probability t_q_x = 1 - t_p_x gives the probability of death within t years. The deferred mortality probability t|u_q_x represents the probability of surviving t years but dying within the following u years: t|u_q_x = t_p_x - (t+u)_p_x = t_p_x * u_q_{x+t}. This probability is important for deferred insurance products and for computing term insurance APVs as sums of deferred one-year death probabilities: A_{x:n}^1 = sum from k=0 to n-1 of v^{k+1} * k|1_q_x.
For example, using a life table with l_{40} = 95000, l_{43} = 93500, and l_{45} = 92000: 3_p_{40} = 93500/95000 = 0.9842, 5_q_{40} = 1 - 92000/95000 = 0.0316, and 3|2_q_{40} = 3_p_{40} - 5_p_{40} = 0.9842 - 0.9684 = 0.0158.
The Survival Function
The survival function S(x) = P(T_0 > x) gives the probability that a newborn survives to at least age x. It is a decreasing function starting at S(0) = 1 and eventually reaching S(omega) = 0, where omega is the limiting age of the mortality table. The relationship to life table functions is S(x) = l_x / l_0.
The future lifetime random variable T_x represents the remaining lifetime of a person currently aged x. Its survival function is P(T_x > t) = t_p_x = S(x+t)/S(x). The density of T_x is f_{T_x}(t) = t_p_x * mu_{x+t}, where mu is the force of mortality. The curtate future lifetime K_x = floor(T_x) is the integer part of the remaining lifetime, representing the number of complete years survived. Its PMF is P(K_x = k) = k_p_x * q_{x+k} = k|1_q_x.
Force of Mortality
The force of mortality mu_x is the instantaneous rate of mortality at exact age x. It is defined as mu_x = -S'(x)/S(x) = -d/dx ln(S(x)). Intuitively, mu_x * dt approximates the probability of dying in the infinitesimal interval (x, x+dt) given survival to age x. Unlike q_x (which is a probability bounded between 0 and 1), mu_x is a rate that can exceed 1 at very old ages.
The force of mortality connects to discrete mortality through the integral relationship: t_p_x = exp(-integral from 0 to t of mu_{x+s} ds). For the special case t = 1: p_x = exp(-integral from 0 to 1 of mu_{x+s} ds). If the force of mortality is constant at mu over an age interval, then t_p_x = exp(-mu*t) within that interval, and q_x = 1 - exp(-mu).
The complete expectation of life at age x is the mean of T_x: e_x_circle = E[T_x] = integral from 0 to infinity of t_p_x dt. The curtate expectation of life is e_x = E[K_x] = sum from k=1 to infinity of k_p_x. The relationship between them depends on the fractional age assumption used, but under UDD, e_x_circle is approximately e_x + 0.5.
Parametric Mortality Laws
Several parametric models describe how the force of mortality varies with age. These provide smooth, continuous mortality functions useful for analytical calculations and for interpolation between tabulated ages.
De Moivre's law: mu_x = 1/(omega - x) for 0 <= x < omega. This produces a uniform distribution of death, with S(x) = (omega - x)/omega. It is unrealistic but simple and sometimes used for basic examples.
Gompertz's law: mu_x = B*c^x for B > 0 and c > 1. This exponential growth in mortality captures the observation that human mortality rates roughly double every 8 to 10 years after middle age. The survival function is S(x) = exp(-B*(c^x - 1)/ln(c)).
Makeham's law: mu_x = A + B*c^x, adding a constant A to the Gompertz model. The constant term represents accident-related mortality that does not depend on age, while the exponential term captures aging. This provides a better fit to human mortality across a wider age range. Under Makeham's law, S(x) = exp(-A*x - B*(c^x - 1)/ln(c)).
Common Pitfalls
Students frequently confuse t_q_x (probability of dying within t years) with q_{x+t} (probability of dying in the one-year interval from age x+t to x+t+1). The first is a cumulative probability; the second is a conditional one-year probability. Another common error is mishandling the deferred mortality probability notation. Remember that t|u_q_x requires surviving t years AND then dying within the next u years.
When working with the force of mortality, be careful about the distinction between mu_x (a rate, not a probability) and q_x (a probability). The approximation q_x is approximately mu_x holds only when mu_x is small. For large values of mu_x at old ages, using the exact relationship p_x = exp(-integral of mu) is essential.
Study Tips
Exam LTAM requires absolute fluency with life table notation and relationships. Practice converting between l_x, d_x, q_x, p_x, and mu_x until these conversions are automatic. Work through problems that require computing multi-year probabilities from life table excerpts, and practice deriving survival functions from parametric mortality laws. Build a reference card with the key formulas and relationships, including the connections between discrete and continuous quantities. The time invested in mastering this notation pays dividends throughout the entire LTAM syllabus, as every subsequent topic builds on these fundamentals.